[Paper Review] Causal Inference under Network Interference Using a Mixture of Randomized Experiments

Yiming Jiang, He Wang|arXiv (Cornell University)|Aug 31, 2023

Advanced Causal Inference TechniquesMathematics3 citations

TL;DR

This paper proposes a mixed randomization design that combines Bernoulli and cluster-based randomization to enable unbiased estimation of average treatment effects under network interference, where units influence each other. It establishes variance bounds of $O(d^2 n^{-1} p^{-1})$ and a matching lower bound of $\ heta(d^{1.5} n^{-1} p^{-1})$, with improved bounds for sparse networks and a weight-invariant design for unknown interference weights.

ABSTRACT

In randomized experiments, the classic Stable Unit Treatment Value Assumption (SUTVA) posits that the outcome for one experimental unit is unaffected by the treatment assignments of other units. However, this assumption is frequently violated in settings such as online marketplaces and social networks, where interference between units is common. We address the estimation of the total treatment effect in a network interference model by employing a mixed randomization design that combines two widely used experimental methods: Bernoulli randomization, where treatment is assigned independently to each unit, and cluster-based randomization, where treatment is assigned at the aggregate level. The mixed randomization design simultaneously incorporates both methods, thereby mitigating the bias present in cluster-based designs. We propose an unbiased estimator for the total treatment effect under this mixed design and show that its variance is bounded by $O(d^2 n^{-1} p^{-1} (1-p)^{-1})$, where $d$ is the maximum degree of the network, $n$ is the network size, and $p$ is the treatment probability. Additionally, we establish a lower bound of $Ω(d^{1.5} n^{-1} p^{-1} (1-p)^{-1})$ for the variance of any mixed design. Moreover, when the interference weights on the network's edges are unknown, we propose a weight-invariant design that achieves a variance bound of $O(d^3 n^{-1} p^{-1} (1-p)^{-1})$, which is aligned with the estimator introduced by Cortez-Rodriguez et al. (2023) under similar conditions.

Motivation & Objective

• To address the violation of the Stable Unit Treatment Value Assumption (SUTVA) in networked experimental settings where units interfere with each other.
• To develop a unified experimental design that combines Bernoulli and cluster-based randomization to better capture interference effects.
• To provide an unbiased estimator for the average treatment effect (ATE) under this mixed design.
• To derive tight variance bounds for the ATE estimator under various network structures and interference assumptions.
• To establish theoretical limits on estimation variance, including a lower bound and improved upper bounds for sparse networks.

Proposed method

• Introduces a mixed randomization design where each unit is independently assigned to either a Bernoulli or cluster-based randomization with a fixed probability.
• Uses a network representation $G(V,E)$ where edges represent interference, with directed edge weights $v_{ij}$ indicating the magnitude of interference from unit $j$ to $i$.
• Proposes an unbiased estimator for the ATE that accounts for both direct and indirect treatment effects via interference weights.
• Applies a graph decomposition algorithm to partition the network into clusters using maximum matchings, ensuring each edge is covered exactly once.
• Derives variance bounds using concentration inequalities and combinatorial arguments, leveraging the maximum degree $d$ and treatment probability $p$.
• Introduces a weight-invariant design that maintains variance bounds even when interference weights are unknown, achieving $O(d^3 n^{-1} p^{-1})$ variance.

Figure 1: Examples of RGG networks: the left figure is randomly generated from $(n,r_{0},r_{1})=(100,10,0)$ and the right figure $(n,r_{0},r_{1})=(100,5,5)$ .

Experimental results

Research questions

• RQ1 Can a mixed design combining Bernoulli and cluster-based randomization yield an unbiased estimator for the average treatment effect under network interference?
• RQ2 What is the theoretical variance bound of the ATE estimator under such a mixed design, and how does it scale with network size $n$, maximum degree $d$, and treatment probability $p$?
• RQ3 Is there a fundamental lower bound on the variance of any mixed design under network interference?
• RQ4 How do the variance bounds improve for sparse networks characterized by a growth constant $\kappa \leq d$?
• RQ5 Can a design be constructed that maintains low variance even when interference weights are unknown?

Key findings

• The proposed estimator for the average treatment effect is unbiased under the mixed randomization design, even in the presence of network interference.
• The variance of the estimator is bounded above by $O(d^2 n^{-1} p^{-1})$, where $d$ is the maximum degree, $n$ is the network size, and $p$ is the treatment probability.
• A matching lower bound of $\Omega(d^{1.5} n^{-1} p^{-1})$ is established, showing the upper bound is nearly tight.
• For sparse networks with growth constant $\kappa \leq d$, the variance bound is improved to $O(\kappa^7 d \, n^{-1} p^{-1})$, reflecting better performance in sparse settings.
• A weight-invariant design is proposed that achieves a variance bound of $O(d^3 n^{-1} p^{-1})$ when interference weights are unknown, ensuring robustness.
• The graph decomposition algorithm ensures that any graph with maximum degree $d$ can be partitioned into at most $2d$ matchings, enabling efficient cluster construction and theoretical analysis.